I am a computer scientist, and thus my expertise is Computer Science.  More precisely, it is Computer Science manifested in its Theory, Practice and Education.

“Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers.” – Wikipedia

Computer science is the discipline of handling information with the use of computational systems.  As a computer scientist, I devote myself to the theory of computation, the practice of designing and implementing computation, and the education of computational thinking as a fundamental scientific paradigm for school students and teachers.

Computer Science: Theory, Practice and Education

Studying the theoretical aspects of Computer Science, which are essentially the mathematical structures underlying computations, e.g., domain theory, topology, category theory, semantics of programming languages, program logic, models of computation.

Applications of Computer Science, e.g., building algorithms, implementations of theoretic computations in real-world situations, simulating real-world systems using computational models.

Computer Science education, e.g., teaching computational thinking & problem solving, and coding to mathematics students at school, professional development of mathematics teachers in the area of computational thinking & problem solving, implementation of computer and video technology in the mathematics classroom.

Research Funding

My research activities would not be possible without the following grants I received from competitive funding past and present, both locally and internationally:

2017-2021:      Computing with Infinite Data (731143-CID)

                        Grant value: SGD 1,522,438 funded by European Union

                        Co-PI with Prof. Dieter Spreen, Siegen University, Germany

                        Leader of Singapore team from NTU

2017-2019:      Modern links with topology and order structures via domains

                        Grant value: SGD 41,200 funded by NIE Academic Research Fund Tier 1

                        Collaborator with A/P Zhao Dongsheng, NIE/NTU, Singapore

2011-2013:     Design and implementation of an exact real and complex scientific calculator in HASKELL

                        Grant value: SGD 50,000 funded by NTU Academic Research Fund Tier 1

                        PI; my co-PIs were A/P Ang Keng Cheng and A/P Zhao Domgsheng

Computer Science: Theory

My main research interest lies in domain theory and its applications.  Domain theory is multi-faceted and inter-disciplinary by nature.  Rough speaking, domains are partially ordered sets endowed with a topology so that it is possible to model the phenomenon of approximation in two ways: order-theoretically and topologically.  Thus, domain theory involves a strong interplay of order and topology.

My favourite quotation that encapsulates succinctly what domain theory is about and the spirit by which research in domain theory should be carried out comes from Jean Goubault-Larrecq, the author of the lovely 2013 book “Non-Hausdorff Topology and Domain theory”:

“Domain theory … is topology done right.” – J. Goubault-Larrecq.

In what follows, I have listed a few areas of domain theory which I have been working at.  These topics form nothing but a tip of the huge iceberg, and certainly do not encompass all of domain theory.  Prospective Ph.D./M.Sc. students who are interested in any of these areas (or, better still, have some new ideas) may drop me an email.   

Scott topological representation theory.  In domain theory, the most prominently featured topology is the Scott topology.  Along the line of connecting order with topology, one central business in domain theory is to understand how much topological information of the Scott space is encoded by its Scott closed set lattice.  More precisely, which distributive complete lattices are exactly the Scott closed set lattice of some poset.  This area of research is known as Scott topological representation theory.  This topic is part of a bigger field of investigation called Stone duality.  I am interested in solving the open problem of giving an order-theoretic characterization of Scott closed set lattices, and other related problems concerning Stone duality arising from Scott topology.

Operational domain theory of programming languages.  Domain theory is actively employed to manufacture denotational semantics for programming languages – a revolutionary discovery made by Dana Scott in the early 1970s, together with Christopher Strachey.  The Scott-Strachey approach interprets the meaning (semantics) of programming language (syntax) using the mathematical model of domains.  More specifically, each data type has its denotation as a domain which is a special type of posets that admits the notion of approximation through its Scott topology.  While the Scott-Strachey approach is powerful in reasoning about program behaviours using the mathematics of domain theory, an inherent mismatch that occurs between the syntax and the semantics plagues the Scott domain model – this problem is known as the full abstraction problem in the literature.   I am interested in working out the operational methods in different programming language environments for reasoning about programs without reliance on any denotational model.

Development of Domain Theory in T₀ spaces.  In the Sixth International Symposium in Domain Theory, Changsha, China, Jimmie Lawson proposed a scientific programme of developing the core of domain theory in the ambient environment of T₀ spaces.  Many generalizations of domain-theoretical results to the T₀ space-setting exist.  For example, (1) corresponding to the dcpo-completion of posets there is a D-completion of topological spaces to d-space; (2) there is a topological version of the order-theoretic Rudin’s lemma; (3)  a topological upgrading of the Scott topology called the SI-topology also exists.   I have responded to Lawson’s call, together with my colleague Zhao Dongsheng, by inventing a novel heuristic now called the “Zhao-Ho replacement principle” to systematically translate known theorem statements of domain-theoretic results to theorem statements that make sense in T₀ spaces.  The job is to try and establish these newly formed topological conjectures; when established, such results will automatically be generalization of known domain-theoretic results.  We call this area of research “Domain Theory in situ T₀ spaces”.  I am interested in investigating which domain-theoretic results may be ‘lifted’ up to the more general realm of T₀ spaces.

Quantitative domain theory.  In Scott’s domain-theoretic semantics for programming languages, the least fixed point of a functional, defined on a domain of Scott continuous functions and with range in the same set, is obtained as a topological limit. The topology involved is precisely the Scott topology. Another way to manufacturing a denotational semantics was proposed by the Dutch school at CWI and originated in work by De Bakker, Zucker and America that made essential use of metric spaces.  Notably, other metric space models for PCF have also been independently considered by Escardó.  To unify the order-theoretic and the metric space approaches, Smyth from Imperial College exploited tools known to non-symmetric topologists, i.e., the notion of quasi-metrics, which essentially are structures that are just short of being a metric space because of the removal of the symmetry axiom.    In other words, for a quasi-metric space, the distance from a given point x to another y may not be equal to that from y to x.   Crucially, every partially ordered set can be viewed as a special quasi-metric space by using the obvious 0-1 encoding, i.e.., the distance from x to y is 0 if x £ y; and 1 otherwise.  Later developments include domain theoretic approaches to dataflow networks, logic programming, domain theoretic approaches to integration, models for probabilistic languages as well as models which incorporate complexity analysis.   Most of these applications involve real number valuation, and hence the adjective ‘quantitative’ as opposed to the adjective ‘qualitative’ which qualifies the traditional order-theoretic approach.  In quantitative domain theory, one important object of study is that of formal balls, i.e., a formal ball is just an abstract pairing of a point x from the quasi-metric space and a non-negative real number r, denoted by B(x,r).  Heckmann and Edalat introduced a certain partial-ordering of formal balls that reflects the containment of metrically open balls.  I am interested in the relationship between the order-theoretic properties of the poset of formal balls and the topological properties of the quasi-metric space.  One central example that illustrates this type of relationship is the Kostanek-Waskiewicz theorem which states that the poset of formal balls arising from a quasi-metric space is a dcpo if and only if its underlying space is Yoneda complete.

Function spaces of domains and topological spaces.  Function spaces of domains are fundamental in domain theory because they allow the modelling of function type programs in functional programming languages.  One specific area of interest regarding function spaces is centred on the question:  Which classes A of topological spaces should be paired up with which classes B of domains so that the Isbell (which is the intrinsic topology of function spaces) and Scott (which is the intrinsic topology when it comes to domains) topologies on function spaces  coincide for any A in A and B in B?

Convergence structures.  Convergence structures are mathematical structures that generalize topological spaces.  There are two advantages working with convergence structures.  Firstly, every instance of a given convergence structure on an underlying space X is a pair comprising a net  (which is intended to be an abstract sequence) and a point x in X that satisfies the constant net and the subnet axioms.  Thus, convergence structures allow one to capture the phenomenon of limits directly with no reference to the notion of neighbourhoods.  Secondly, the category of convergence structures is Cartesian closed, which is a very pleasing property for modelling programs of functional programming languages – one which the category of topological spaces does not possess.  To add on, a significant result by D. S. Scott is that the lim-inf convergence on a dcpo is topological if and only if the dcpo is continuous, and this result highlights the significance of domains in relation to convergence structures.  I am interested in the study of various convergence relations that can be defined on domains or domain-like spaces, and the characterizations of domains (or spaces) for which such convergence structures are topological.

Computer Science: Practice

I am keenly interested in actual implementations of theory of computation in real-world systems.  Two particular applications are of particular interest to me: Exact Real Calculator for Everyone and Artificial Life. Prospective Ph.D./M.Sc. students who have interest in any of these implementations may contact me via the above email.

Exact Real Calculator for Everyone.  Current computers handle real number computations using the well-known floating point system (FPS, for short).  Recognized for its simplicity and versatility, FPS even has its own IEEE standard (IEEE 754).  However, FPS has a serious shortcoming, numbers with too many (or infinitely) significant digits can only be represented by nearby rational numbers, and this produces round-off errors. Though such inaccuracies are generally tolerable, there are inevitable exceptions. For example, functions (such as the logistic map) which are sensitive to initial input can yield behemoth deviations even when seemingly negligible errors occur in the input. 

To tackle this real-world problem, I have developed an exact real arithmetic calculator which is able to deal with all pre-calculus operations/functions – ERCE, Exact Real Calculator for Everyone, which has been implemented in HASKELL, a functional programming language.  The representation we use for real numbers is signed digit representation, and the real-exact arithmetic programs rely on list types – a special recursive types.  The theory of recursive types had earlier been given a comprehensive treatment in my 2014 paper “An operational domain-theoretic treatment of recursive types”.  The practical implementation of this theory is manifested in the physical calculator ERCE, which is supported by Graphic User Interface and can be used by anyone without any prior knowledge in programming and mathematics (e.g.. even primary school children). This implementation ERCE is significant as far as technological advancement is concerned.  The ERCE package can be downloaded here

Artificial Life.  As an application of category theory (and domain theory), some joint works with Dr. Dominique Chu (senior lecturer at the University of Kent) performed during my early research career showcase the interdisciplinary connection of my area of expertise with Computational Biology, especially in Artificial Life.   

Computer Science: Education

As a computer scientist working in National Institute of Education – the hub of education in Singapore, I am naturally interested in educational issues related to computer science.  But I am a mathematician too, and so I see myself contributing to Mathematics Education just as well.  Prospective M.A. students who have interest in any of the following research topics/school-based projects are welcome to drop me an email.

Computer Science Education

Many developed countries, including Singapore, have identified Coding or Computer Programming as one of the essential 21^st century competencies. In his official speech at Smart Nation launch on 24 November 2014, Singapore Prime Minister Lee Hsien Loong envisioned a Smart Nation as one “where people live meaningful and fulfilled lives, enabled seamlessly by technology, offering exciting opportunities for all. ... where networks of sensors and smart devices enable us to live sustainably and comfortably”.  In the same speech, schools were specifically urged to expose students to Information Technology and basic coding skills.  Here, we are talking about teaching Singaporean children the new discipline of computing.

I am of the opinion that learning a discipline is a life-long activity, and it requires a very distinctive mindset.  This mindset which Jeanette Wing called Computational Thinking is a paradigm of attacking complex problems by

• breaking these down into smaller tasks,
• finding representations for them so as to make them tractable, and in most cases,
• creating efficient algorithms to solve them.

Computational thinking and coding in schools.  In the 21^st century, computation must be seen as one of the pillars of science, together with theory and experimentation – the big idea has already worked out by Jeannette Wing some time ago.  What remains to be done is the unglamorous job of working out in concrete terms the educational aspect of this philosophy.  How can I as an educator:

• promote computational thinking in schools?
• teach students computational thinking?
• equip teachers with the professional capacity to teach computational thinking?

I am interested in both research and practice that in one way or another answer the above questions. 

Since 2013, I began to convince both researchers and practitioners in the Asian Technology Conference in Mathematics about the importance of the use of computer technology and coding in mathematics education.

In 2017, I started work with the mathematics teachers of Anderson Secondary School on how to harness computational thinking in teaching and learning secondary school mathematics throughout all levels.  This work begins with the identification of the subtopics, at each level, for which computational aspects can be exploited to deepen the learning of concepts, techniques and problem solving disposition.  Once these subtopics and their corresponding computational elements have been identified, a teaching innovation that makes essential use of computer technology

Mathematics Education

Working in the Mathematics and Mathematics Education (MME) Academic Group in NIE provides me with many collaborative opportunities with my NIE colleagues in the area of mathematics education.  I see my research efforts in mathematics education as augmentation to my scope of research in computer science education.

Video-technology in mathematics education.  Grounded image theory proposes the affordance that teaching moments can be captured on video and replayed later, as many times as one wishes, should be harnessed to enhance teacher training in methods courses.  Video technology is also fundamental in realizing the “flipped” part of the flipped classroom pedagogy so that traditional face-to-face information transmission can be “displaced” or “inverted” before classroom time.  My research interest is to investigate how modern computer technology, especially online learning management systems such as LAMS, can work hand-in-hand with video-technology to make teaching and learning more effective.

Tertiary mathematics education.  Literature in tertiary mathematics education reveals many problems encountered by students as they transit from high-school mathematics to university mathematics.  In particular, students feel that the high school mathematics they have learnt seem to be irrelevant because there is such a huge jump in the level of rigor required of them in tertiary mathematics.  This jump is termed as the first “discontinuity” by Felix Klein.  For mathematics majors who plan to become teachers, there will be a second “discontinuity” in the sense that what they learnt in the university about mathematics do not seem to help them in teaching school mathematics, and so they quickly revert to the same old manner in which their mathematics teachers taught.  MME, as an AG, is actively solving this problem of double-discontinuity by relying on Ralph Tyler’s model of curriculum design based on objectives.  My research interest is to investigate the various problems associated to transition from high school to university mathematics, and how such issues can be addressed via a change in curriculum orientation.  I am also interested in understanding the effect of the curricular changes in ‘A’ Level Mathematics on the teaching and learning of mathematics at the tertiary level.

Problem solving.  Problem solving is the central theme in the Singapore Mathematics Curriculum, i.e., it is right at the hub of the pentagonal wheel in the Singapore Mathematics Framework advocated by the Ministry of Education, Singapore.  I am a true believer of using problem solving as a fundamental approach to teaching and learning mathematics.  Notably, my MME colleagues from the MProSE (Mathematics Problem Solving for Everyone) and MInD (Mathematics problem solving: Infusion and Diffusion) teams have done impactful works in diffusing and infusing of the problem solving paradigm to several pilot schools.  I am keenly interested in issues connected to the sustainability and scaling-up of the “Mathematics Problem Solving for Everyone”.

Mathematical modelling.  Computational thinking and mathematical modelling go hand-in-hand.  Mathematical modelling provides the platform by which students can experience first-hand how their school mathematics can be applied to solve problems in real-world situations.  My NIE colleagues, e.g., Ang Keng Cheng, Soon Wan Mei, have also done the hard work of building up the framework for professional development for teachers to teach mathematical modelling to students at school.  My research interest lies in the connection of computational thinking with mathematical modelling.  More precisely, I am keen in finding ways by which computer technology and coding skills can be applied to “put life into” the mathematical model which in turn yields solutions to real-life problems.